Subgroups — #13083697

Order 1440.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₁₈₀	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₃₆	C₂	1	1	yes	maximal, characteristic
3	C₉×#672	C₂	1	1	yes	maximal, characteristic
4	#90948	—	3	1	no	maximal
5	Dic₉₀	—	2	1	no	—
6	C₅×Dic₁₈	—	2	1	no	—
7	C₅×C₈×C₉	C₂²	1	1	yes	characteristic
8	Dic₇₂	—	5	1	no	maximal, Hall
9	Dic₆₀	—	3	1	no	—
10	C₅×Dic₁₂	—	3	1	no	—
11	C₃×#672	D₃	1	1	yes	characteristic
12	C₄×C₅×C₉	D₄	1	1	yes	characteristic
13	Dic₄₅	—	4	1	no	—
14	C₅×Dic₉	—	4	1	no	—
15	#3961	—	9	1	no	Hall
16	Dic₃₆	—	5	1	no	—
17	Dic₃₆	D₅	1	1	yes	characteristic
18	C₉×C₁₆	—	5	1	no	—
19	Dic₃₀	—	6	1	no	—
20	C₅×Dic₆	—	6	1	no	—
21	C₃×C₅×C₈	C₂×D₃	1	1	yes	characteristic
22	Dic₂₄	—	15	1	no	—
23	C₂×C₅×C₉	D₈	1	1	yes	characteristic
24	C₅×Q₁₆	—	9	1	no	—
25	Dic₂₀	—	9	1	no	—
26	#672	D₉	1	1	yes	characteristic
27	Dic₁₈	—	2	1	no	—
28	Dic₁₈	—	10	1	no	—
29	C₈×C₉	C₂×D₅	1	1	yes	characteristic
30	C₃×C₄×C₅	D₁₂	1	1	yes	characteristic
31	Dic₁₅	—	12	1	no	—
32	C₅×Dic₃	—	12	1	no	—
33	Dic₁₂	—	3	1	no	—
34	Dic₁₂	—	15	1	no	—
35	C₃×C₁₆	—	5	1	no	—
36	C₅×C₉	Q₃₂	1	1	yes	characteristic, complemented, Hall
37	C₅×Q₈	—	18	1	no	—
38	Dic₁₀	—	18	1	no	—
39	C₅×C₈	C₂×D₉	1	1	yes	characteristic
40	C₄×C₉	C₂²⋊D₅	1	1	yes	characteristic
41	Dic₉	—	4	1	no	—
42	Dic₉	—	20	1	no	—
43	Q₃₂	—	45	1	no	Sylow
44	C₂×C₃×C₅	D₂₄	1	1	yes	characteristic
45	Dic₆	—	6	1	no	—
46	Dic₆	—	30	1	no	—
47	C₃×C₈	D₃×D₅	1	1	yes	characteristic
48	C₄×C₅	—	36	1	no	—
49	C₄×C₅	D₃₆	1	1	yes	characteristic
50	Dic₅	—	36	1	no	—
51	C₂×C₉	#686	1	1	yes	characteristic
52	Q₁₆	—	9	1	no	—
53	Q₁₆	—	45	1	no	—
54	C₁₆	—	5	1	no	—
55	C₃×C₅	Dic₂₄	1	1	yes	characteristic
56	C₃×C₄	#1203	1	1	yes	characteristic
57	Dic₃	—	12	1	no	—
58	Dic₃	—	60	1	no	—
59	C₂×C₅	D₇₂	1	1	yes	characteristic
60	C₉	#3961	1	1	yes	characteristic, complemented, Sylow
61	Q₈	—	18	1	no	—
62	Q₈	—	90	1	no	—
63	C₈	D₅×D₉	1	1	yes	characteristic
64	C₂×C₃	#6654	1	1	yes	characteristic
65	C₅	Dic₇₂	1	1	yes	characteristic, complemented, Sylow
66	C₄	—	36	1	no	—
67	C₄	—	180	1	no	—
68	C₄	#67080	1	1	yes	characteristic
69	C₃	#90948	1	1	yes	characteristic
70	C₂	#10623544	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10623544
commutator, Fitting, cosocle	C₅×C₈×C₉	C₂²
Frattini	C₃×C₈	D₃×D₅
socle, prime radical	C₂×C₃×C₅	D₂₄
hypercenter	C₈	D₅×D₉
hypocenter	C₅×C₉	Q₃₂
LCS	G ▷ C₅×C₈×C₉ ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
36. C₅×C₉	Q₃₂	semidirect
60. C₉	#3961	semidirect
65. C₅	Dic₇₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Dic₁₈₀	2
C₅×Dic₃₆	2
C₉×#672	2
#90948	3
Dic₇₂	5